#8651424

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,13,14)(11,15,19,17)(12,16,21,22)(18,23,20,24)(25,26)(27,38)(28,35)(29,34)(30,39)(31,33)(32,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,27,32,31)(26,33,37,38)(28,39,29,36)(30,35,40,34);(2,3)(6,7)(10,14)(15,17)(16,22)(23,24)(25,28,32,29)(26,34,37,35)(27,39,31,36)(30,38,40,33);(4,8)(6,7)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,29,32,28)(26,35,37,34)(27,39,31,36)(30,38,40,33);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,30,32,40)(26,36,37,39)(27,34,31,35)(28,33,29,38);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,31,32,27)(26,33,37,38)(28,36,29,39)(30,35,40,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8558620);